MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issubrg2 Structured version   Visualization version   GIF version

Theorem issubrg2 20609
Description: Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
Hypotheses
Ref Expression
issubrg2.b 𝐵 = (Base‘𝑅)
issubrg2.o 1 = (1r𝑅)
issubrg2.t · = (.r𝑅)
Assertion
Ref Expression
issubrg2 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑅,𝑦   𝑥, · ,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   1 (𝑥,𝑦)

Proof of Theorem issubrg2
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subrgsubg 20594 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
2 issubrg2.o . . . 4 1 = (1r𝑅)
32subrg1cl 20597 . . 3 (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
4 issubrg2.t . . . . . 6 · = (.r𝑅)
54subrgmcl 20601 . . . . 5 ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑥𝐴𝑦𝐴) → (𝑥 · 𝑦) ∈ 𝐴)
653expb 1119 . . . 4 ((𝐴 ∈ (SubRing‘𝑅) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥 · 𝑦) ∈ 𝐴)
76ralrimivva 3200 . . 3 (𝐴 ∈ (SubRing‘𝑅) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
81, 3, 73jca 1127 . 2 (𝐴 ∈ (SubRing‘𝑅) → (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴))
9 simpl 482 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝑅 ∈ Ring)
10 simpr1 1193 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubGrp‘𝑅))
11 eqid 2735 . . . . . . 7 (𝑅s 𝐴) = (𝑅s 𝐴)
1211subgbas 19161 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴 = (Base‘(𝑅s 𝐴)))
1310, 12syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 = (Base‘(𝑅s 𝐴)))
14 eqid 2735 . . . . . . 7 (+g𝑅) = (+g𝑅)
1511, 14ressplusg 17336 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1610, 15syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (+g𝑅) = (+g‘(𝑅s 𝐴)))
1711, 4ressmulr 17353 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → · = (.r‘(𝑅s 𝐴)))
1810, 17syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → · = (.r‘(𝑅s 𝐴)))
1911subggrp 19160 . . . . . 6 (𝐴 ∈ (SubGrp‘𝑅) → (𝑅s 𝐴) ∈ Grp)
2010, 19syl 17 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Grp)
21 simpr3 1195 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)
22 oveq1 7438 . . . . . . . . 9 (𝑥 = 𝑢 → (𝑥 · 𝑦) = (𝑢 · 𝑦))
2322eleq1d 2824 . . . . . . . 8 (𝑥 = 𝑢 → ((𝑥 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑦) ∈ 𝐴))
24 oveq2 7439 . . . . . . . . 9 (𝑦 = 𝑣 → (𝑢 · 𝑦) = (𝑢 · 𝑣))
2524eleq1d 2824 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑢 · 𝑦) ∈ 𝐴 ↔ (𝑢 · 𝑣) ∈ 𝐴))
2623, 25rspc2v 3633 . . . . . . 7 ((𝑢𝐴𝑣𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴 → (𝑢 · 𝑣) ∈ 𝐴))
2721, 26syl5com 31 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴))
28273impib 1115 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴𝑣𝐴) → (𝑢 · 𝑣) ∈ 𝐴)
29 issubrg2.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
3029subgss 19158 . . . . . . . . . 10 (𝐴 ∈ (SubGrp‘𝑅) → 𝐴𝐵)
3110, 30syl 17 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴𝐵)
3231sseld 3994 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑢𝐴𝑢𝐵))
3331sseld 3994 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑣𝐴𝑣𝐵))
3431sseld 3994 . . . . . . . 8 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑤𝐴𝑤𝐵))
3532, 33, 343anim123d 1442 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → ((𝑢𝐴𝑣𝐴𝑤𝐴) → (𝑢𝐵𝑣𝐵𝑤𝐵)))
3635imp 406 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢𝐵𝑣𝐵𝑤𝐵))
3729, 4ringass 20271 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3837adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
3936, 38syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢 · 𝑣) · 𝑤) = (𝑢 · (𝑣 · 𝑤)))
4029, 14, 4ringdi 20278 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4140adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4236, 41syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → (𝑢 · (𝑣(+g𝑅)𝑤)) = ((𝑢 · 𝑣)(+g𝑅)(𝑢 · 𝑤)))
4329, 14, 4ringdir 20279 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4443adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐵𝑣𝐵𝑤𝐵)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
4536, 44syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ (𝑢𝐴𝑣𝐴𝑤𝐴)) → ((𝑢(+g𝑅)𝑣) · 𝑤) = ((𝑢 · 𝑤)(+g𝑅)(𝑣 · 𝑤)))
46 simpr2 1194 . . . . 5 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 1𝐴)
4732imp 406 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → 𝑢𝐵)
4829, 4, 2ringlidm 20283 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
4948adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → ( 1 · 𝑢) = 𝑢)
5047, 49syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → ( 1 · 𝑢) = 𝑢)
5129, 4, 2ringridm 20284 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5251adantlr 715 . . . . . 6 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐵) → (𝑢 · 1 ) = 𝑢)
5347, 52syldan 591 . . . . 5 (((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) ∧ 𝑢𝐴) → (𝑢 · 1 ) = 𝑢)
5413, 16, 18, 20, 28, 39, 42, 45, 46, 50, 53isringd 20305 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝑅s 𝐴) ∈ Ring)
5531, 46jca 511 . . . 4 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → (𝐴𝐵1𝐴))
5629, 2issubrg 20588 . . . 4 (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
579, 54, 55, 56syl21anbrc 1343 . . 3 ((𝑅 ∈ Ring ∧ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)) → 𝐴 ∈ (SubRing‘𝑅))
5857ex 412 . 2 (𝑅 ∈ Ring → ((𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴) → 𝐴 ∈ (SubRing‘𝑅)))
598, 58impbid2 226 1 (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  s cress 17274  +gcplusg 17298  .rcmulr 17299  Grpcgrp 18964  SubGrpcsubg 19151  1rcur 20199  Ringcrg 20251  SubRingcsubrg 20586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrng 20563  df-subrg 20587
This theorem is referenced by:  opprsubrg  20610  subrgint  20612  issubrg3  20617  issubrgd  21214  cnsubrglem  21452  cnsubrglemOLD  21453  mplsubrg  22043  mplind  22112  dmatsrng  22523  scmatsrng  22542  scmatsrng1  22545  cpmatsrgpmat  22743  elrgspnlem2  33233
  Copyright terms: Public domain W3C validator